10edo - Revision history

Lériendil: /* Intervals and notation */

2026-02-26T21:00:24Z

Intervals and notation

← Older revision		Revision as of 21:00, 26 February 2026
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	=== Intervals and notation ===		=== Intervals and notation ===
	~~[TBD]~~		10edo collapses the diatonic scale into 5edo; in that way, B and C are the same note, and so are E and F. Though, to notate the other five notes of 10edo, we could simply redefine "sharps" and "flats" to correspond to steps of 10edo, we elect to use the symbols "up" and "down" for this purpose.

			Another notation scheme uses 10edo's native heptatonic scale, which is mosh (3L 4s), generated by the neutral third. We can assign the names 1234567 to the notes following the 1-2-1-2-1-2-1 step pattern of the mosh scale (which resembles 12edo major diatonic with two of its intervals shortened). Sharps and flats are then one step of 10edo following the conventional definition of the larger step minus the smaller.

	JI approximations (within the 17-odd-limit) of steps in 10edo, as well as the aforementioned ways of notating 10edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.		JI approximations (within the 17-odd-limit) of steps in 10edo, as well as the aforementioned ways of notating 10edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.

Lériendil: /* Intervals and notation */

2026-02-26T19:32:05Z

Intervals and notation

← Older revision		Revision as of 19:32, 26 February 2026
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	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
	! rowspan="2" \| Edostep !! rowspan="2" \| Cents !! rowspan="2" \| 2.3.5.7.13.17 <br> JI approximation !! colspan="2" \| Notation !! rowspan="2" \| 10-form interval category		! rowspan="2" \| Edostep !! rowspan="2" \| Cents !! rowspan="2" \| 2.3.5.7.13.17 <br> JI approximation !! colspan="2" \| Notation !! rowspan="2" \| 10-form <br> interval category
	\|-		\|-
	! rowspan="1" \| Pentatonic !! rowspan="1" \| Mosh		! rowspan="1" \| Pentatonic !! rowspan="1" \| Mosh

Lériendil: /* Intervals and notation */

2026-02-26T19:31:52Z

Intervals and notation

← Older revision		Revision as of 19:31, 26 February 2026
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	\|0		\|0
	\|1/1		\|1/1
	\|C		\|B, C
	\|1		\|1
	\|Unison		\|Unison
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	\|120		\|120
	\|'''17/16''', 16/15, [15/14], 14/13, 13/12		\|'''17/16''', 16/15, [15/14], 14/13, 13/12
	\|		\|^B, ^C, vD
	\|		\|2
	\|Second		\|Second
	\|-		\|-
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	\|240		\|240
	\|'''9/8''', 8/7, 7/6		\|'''9/8''', 8/7, 7/6
	\|		\|D
	\|		\|2#, 3b
	\|Unilatus		\|Unilatus
	\|-		\|-
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	\|360		\|360
	\|6/5, [16/13], '''5/4'''		\|6/5, [16/13], '''5/4'''
	\|		\|^D, vE, vF
	\|		\|3
	\|Third		\|Third
	\|-		\|-
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	\|480		\|480
	\|9/7, 13/10, 4/3		\|9/7, 13/10, 4/3
	\|		\|E, F
	\|		\|4
	\|Fourth		\|Fourth
	\|-		\|-
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	\|600		\|600
	\|7/5, [17/12], 10/7		\|7/5, [17/12], 10/7
	\|		\|^E, ^F, vG
	\|		\|4#, 5b
	\|Median		\|Median
	\|-		\|-
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	\|720		\|720
	\|'''3/2''', 20/13, 14/9		\|'''3/2''', 20/13, 14/9
	\|		\|G
	\|		\|5
	\|Fifth		\|Fifth
	\|-		\|-
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	\|840		\|840
	\|8/5, ['''13/8'''], 5/3		\|8/5, ['''13/8'''], 5/3
	\|		\|^G, vA
	\|		\|6
	\|Sixth		\|Sixth
	\|-		\|-
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	\|960		\|960
	\|12/7, '''7/4''', 16/9		\|12/7, '''7/4''', 16/9
	\|		\|A
	\|		\|6#, 7b
	\|Antilatus		\|Antilatus
	\|-		\|-
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	\|1080		\|1080
	\|24/13, 13/7, [28/15], '''15/8''', 32/17		\|24/13, 13/7, [28/15], '''15/8''', 32/17
	\|		\|^A, vB, vC
	\|		\|7
	\|Seventh		\|Seventh
	\|-		\|-
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	\|1200		\|1200
	\|2/1		\|2/1
	\|		\|B, C
	\|		\|1
	\|Octave		\|Octave
	\|}		\|}

Lériendil: /* Intervals and notation */

2026-02-26T19:29:27Z

Intervals and notation

← Older revision		Revision as of 19:29, 26 February 2026
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	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
	! rowspan="2" \| Edostep !! rowspan="2" \| Cents !! rowspan="2" \| 2.9.15.7.11.17 <br> JI approximation !! colspan="2" \| Notation !! rowspan="2" \| 10-form interval category		! rowspan="2" \| Edostep !! rowspan="2" \| Cents !! rowspan="2" \| 2.3.5.7.13.17 <br> JI approximation !! colspan="2" \| Notation !! rowspan="2" \| 10-form interval category
	\|-		\|-
	! rowspan="1" \| Pentatonic !! rowspan="1" \| Mosh		! rowspan="1" \| Pentatonic !! rowspan="1" \| Mosh

Lériendil: /* Intervals and notation */

2026-02-26T19:28:37Z

Intervals and notation

← Older revision		Revision as of 19:28, 26 February 2026
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	=== Intervals and notation ===		=== Intervals and notation ===
	~~As 11edo does not have a chain of fifths (unless one counts the hard~~ [~~[2L 5s]~~] ~~scale generated by 6\11) suitable for notation, the task of notating 11edo proves challenging.~~		[TBD]

	One approach is to treat 11edo as a subset of 22edo. The native-fifths notation for 22edo is derived through stacking 22edo's tempered version of 3/2 (13\22) and assigning names accordingly. Only every other position in the chain of fifths is a note of 11edo, so therefore starting from C, the notes C, D, E, Gb, F#, Ab, G#, Bb, and A# are part of 11edo.		JI approximations (within the 17-odd-limit) of steps in 10edo, as well as the aforementioned ways of notating 10edo, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.

	In 22edo, as the diatonic MOS has [[hardness]] 4:1, a sharp corresponds to +3 steps of 22edo while a flat corresponds to -3 (representing the diatonic chroma in each case). In addition, the accidentals ^ and v raise and lower by one step of 22edo, respectively. The motion of one step in 11edo is therefore represented by the combination of the down and sharp accidentals: vC# is the step above C.

	This notation can be seen as rather awkward, however, as it forces inconvenient representations, leaves glaring gaps in its system of nominals, and in general fails to reflect the radically non-diatonic structure of 11edo. Therefore, the other approach is to use a notational scale that is native to 11edo. The most obvious option here is [[smitonic]] (4L 3s), which not only is heptatonic, but is generated by 5/3, arguably 11edo's most important consonance.

	For the nominals of smitonic, we opt to use the numbers 1 through 7, to avoid confusion with diatonic notes; 1 is identified here with C, and 1234567 follow the LsLLsLs mode of smitonic. As the MOS chroma is 1 step of 11edo, sharps and flats simply alter by 1 step in the smitonic-based system.

	~~"Interval categories" are based on 22edo usage of the [[ADIN]] system; "nearminor/major" and "subminor/supermajor" correspond here to simple 5-limit and [[7-limit]] qualities.~~

	JI approximations (within the 17-odd-limit) of steps in ~~11edo~~, as well as the aforementioned ways of notating ~~11edo~~, are detailed in the table below. Intervals within 5 cents are in [brackets], and odd harmonics are bolded.
	{\| class="wikitable"		{\| class="wikitable"
	\|+		\|+
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	\|1		\|1
	\|120		\|120
	\|16/15, [15/14], ~~''10~~/~~9''~~		\|'''17/16''', 16/15, [15/14], 14/13, 13/12
	\|		\|
	\|		\|
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	\|2		\|2
	\|240		\|240
	\|9/8, 8/7, 7/6		\|'''9/8''', 8/7, 7/6
	\|		\|
	\|		\|
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	\|3		\|3
	\|360		\|360
	\|6/5, [16/13], 5/4		\|6/5, [16/13], '''5/4'''
	\|		\|
	\|		\|
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	\|6		\|6
	\|720		\|720
	\|3/2, 20/13, 14/9		\|'''3/2''', 20/13, 14/9
	\|		\|
	\|		\|
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	\|7		\|7
	\|840		\|840
	\|8/5, [13/8], 5/3		\|8/5, ['''13/8'''], 5/3
	\|		\|
	\|		\|
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	\|8		\|8
	\|960		\|960
	\|12/7, 7/4, 16/9		\|12/7, '''7/4''', 16/9
	\|		\|
	\|		\|
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	\|9		\|9
	\|1080		\|1080
	\|~~''9~~/~~5''~~, [28/15], 15/8		\|24/13, 13/7, [28/15], '''15/8''', 32/17
	\|		\|
	\|		\|

Lériendil: /* General theory */

2026-02-26T19:25:44Z

General theory

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 While 10edo has [[consistent]] representations of every interval within the no-11s 17-[[odd-limit]] save [[10/9]] and its complement, it still makes extreme temperings (such as the vanishing of [[25/24]]) that put many of these intervals out of recognition. The primary subgroup in which 10edo is of notable accuracy is 2.7.13.15; in particular, the intervals 16/13 and [[15/14]] are approximated within 0.6{{c}}.
 {{Harmonics in ED|10|19|prime}}
 == Compositional theory ==

Hotcrystal0: adding the edo navbox

2026-02-26T16:01:25Z

adding the edo navbox

← Older revision		Revision as of 16:01, 26 February 2026
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	The 10-form's key features are the presence of the tritone as its own interval category separate from fourths and fifths, and the moving of 9/7 and 7/6 away from the category representing thirds, into the categories corresponding to fourths and major seconds respectively. The interval classes of "second" and "seventh" are split up into correspondents to "minor" and "major". This in effect gives the simplest 7-limit intervals their own pair of categories separate from the simplest 5-limit intervals, much as upgrading from the 5-form to the 7-form gives the simplest 5-limit intervals their own pair of categories.		The 10-form's key features are the presence of the tritone as its own interval category separate from fourths and fifths, and the moving of 9/7 and 7/6 away from the category representing thirds, into the categories corresponding to fourths and major seconds respectively. The interval classes of "second" and "seventh" are split up into correspondents to "minor" and "major". This in effect gives the simplest 7-limit intervals their own pair of categories separate from the simplest 5-limit intervals, much as upgrading from the 5-form to the 7-form gives the simplest 5-limit intervals their own pair of categories.

	{{Cat\|Edos}}		{{Cat\|Edos}}{{Navbox EDO}}

Lériendil at 14:20, 23 February 2026

2026-02-23T14:20:26Z

← Older revision		Revision as of 14:20, 23 February 2026
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	'''10edo''', or 10 equal divisions of the octave (sometimes called '''10-TET''' or '''10-tone equal temperament'''), is the equal tuning featuring steps of (1200/10) = 120 cents, 10 of which stack to the perfect octave [[2/1]].		'''10edo''', or 10 equal divisions of the octave (sometimes called '''10-TET''' or '''10-tone equal temperament'''), is the equal tuning featuring steps of (1200/10) = 120 cents, 10 of which stack to the perfect octave [[2/1]].

	As the double of [[5edo]], 10edo has its sharp [[perfect fifth\|fifth]] of 720{{c}}, and approximates [[7/4]] well by 960{{c}}. It divides this fifth in two so that it possesses a [[neutral third]] of 360{{c}} that is extremely close to [[16/13]]. Both [[6/5]] and [[5/4]] are mapped to this neutral third, and in this fashion 10edo provides a ~~very~~ primitive basis for the [[7-limit]].		As the double of [[5edo]], 10edo has its sharp [[perfect fifth\|fifth]] of 720{{c}}, and approximates [[7/4]] well by 960{{c}}. It divides this fifth in two so that it possesses a [[neutral third]] of 360{{c}} that is extremely close to [[16/13]]. Both [[6/5]] and [[5/4]] are mapped to this neutral third, and in this fashion 10edo provides a primitive structural basis for the [[7-limit]].

	10edo occupies a position intermediate between the "[[albitonic]]" systems such as [[7edo]], and "chromatic" systems such as [[12edo]]. While scales such as [[3L 4s]] can be used as a structural backbone when working within 10edo, the [[10-form]] itself can serve as a basic form in tuning systems, more analogous to the [[7-form]]. In particular, 10edo's division of the perfect fifth into six and the [[perfect fourth]] into four allows it to represent a trivial tuning of temperaments such as [[Miracle]] and [[Negri]].		10edo occupies a position intermediate between the "[[albitonic]]" systems such as [[7edo]], and "chromatic" systems such as [[12edo]]. While scales such as [[3L 4s]] can be used as a structural backbone when working within 10edo, the [[10-form]] itself can serve as a basic form in tuning systems, more analogous to the [[7-form]]. In particular, 10edo's division of the perfect fifth into six and the [[perfect fourth]] into four allows it to represent a trivial tuning of temperaments such as [[Miracle]] and [[Negri]].

Lériendil: /* Chords */

2026-02-23T14:19:13Z

Chords

← Older revision		Revision as of 14:19, 23 February 2026
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	Arto: 0-2-6-(8)		Arto: 0-2-6-(8)

	See also: [[Oneirotonic#~~Chords_of_oneirotonic~~]]		See also: [[Oneirotonic#Chords of oneirotonic]]

	=== Scales ===		=== Scales ===

Lériendil: /* General theory */

2026-02-23T14:15:03Z

General theory

← Older revision		Revision as of 14:15, 23 February 2026
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	While 10edo has [[consistent]] representations of every interval within the no-11s 17-[[odd-limit]] save [[10/9]] and its complement, it still makes extreme temperings (such as the vanishing of [[25/24]]) that put many of these intervals out of recognition. The primary subgroup in which 10edo is of notable accuracy is 2.7.13.15; in particular, the intervals 16/13 and [[15/14]] are approximated within 0.6{{c}}.		While 10edo has [[consistent]] representations of every interval within the no-11s 17-[[odd-limit]] save [[10/9]] and its complement, it still makes extreme temperings (such as the vanishing of [[25/24]]) that put many of these intervals out of recognition. The primary subgroup in which 10edo is of notable accuracy is 2.7.13.15; in particular, the intervals 16/13 and [[15/14]] are approximated within 0.6{{c}}.

	{{Harmonics in ED\|10\|19\|prime}}		{{Harmonics in ED\|10\|19\|prime}}